Your analysis is a reasonable first cut, but the above question is a good one, and your reply is rather incorrect.
First, you haven't really done a Bayesian analysis, for several reasons.
The easiest problem to fix is, you didn't specify any priors. We could validate your style of calculation by assuming single-engine failures are IID with parameter "p," uniform on the unit interval. (A beta distribution would be the standard conjugate prior.)
If we go ahead and make the uniform-p assumption, then what you've calculated is a most-likely posterior value of "p" (35/36). (This is the maximum-likelihood estimate for p.) But in a Bayesian world, "p" has a full posterior distribution, not just a most-likely value.
So, still in the Bayesian world, the probability of failure (P(N_fail > 2)) must be calculated on the basis of not the most likely p, but the posterior of "p". You didn't do this; you just used the most-likely p.
Which brings us to the second problem with your reply. You really can get a confidence value on P(N_fail > 2). You can get a full posterior distribution! It will be a 1-dimensional density on [0,1]. And you could calculate this, either by simulation or by an analytic procedure (because it's a low-dimensional problem).
This posterior on P(N_fail > 2) would be the answer to the parent's question. It would probably be rather "fat", validating the intuition that we don't have much data.
My reply might be a little smarty-pants. Sorry if it is! As I said, your analysis is a reasonable first cut.
The easiest problem to fix is, you didn't specify any priors.
Read the whole thing more carefully. I started with the calculation for the maximum likelihood estimate, but I ended with a prior with equal a priori likelihoods for failure rates 0%, 0.01%, 0.02%, ..., 99.99%, 100%. This is a reasonable discretization of uniform on the unit interval.
My reply might be a little smarty-pants. Sorry if it is!
I would suggest that before indulging a tendency to be a smarty pants, that it is good to read the whole thing.
You're right, the second half of your post in effect puts a two-atom prior on "p", with zero everywhere else, and then goes on to use a sequence of such atoms, which would approximate a uniform prior. It's more standard to use smooth priors, because we don't have precise information, but you are right, I was not reading carefully.
You are still in error that there is not a way to describe the uncertainty in your estimate of the posterior probability of system failure. It has a posterior distribution, like everything else in a Bayesian analysis. You would compute it as I described -- Monte Carlo would be easiest.
First, you haven't really done a Bayesian analysis, for several reasons.
The easiest problem to fix is, you didn't specify any priors. We could validate your style of calculation by assuming single-engine failures are IID with parameter "p," uniform on the unit interval. (A beta distribution would be the standard conjugate prior.)
If we go ahead and make the uniform-p assumption, then what you've calculated is a most-likely posterior value of "p" (35/36). (This is the maximum-likelihood estimate for p.) But in a Bayesian world, "p" has a full posterior distribution, not just a most-likely value.
So, still in the Bayesian world, the probability of failure (P(N_fail > 2)) must be calculated on the basis of not the most likely p, but the posterior of "p". You didn't do this; you just used the most-likely p.
Which brings us to the second problem with your reply. You really can get a confidence value on P(N_fail > 2). You can get a full posterior distribution! It will be a 1-dimensional density on [0,1]. And you could calculate this, either by simulation or by an analytic procedure (because it's a low-dimensional problem).
This posterior on P(N_fail > 2) would be the answer to the parent's question. It would probably be rather "fat", validating the intuition that we don't have much data.
My reply might be a little smarty-pants. Sorry if it is! As I said, your analysis is a reasonable first cut.